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1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
124
Double Weighted Distribution & Double Weighted Exponential
Distribution
Kareema abed al-kadim1*
Ahmad Fadhil Hantoosh2**
1*., 2**
. College of Education of Pure Sciences, University of Babylon/ Hilla
*
E-mail of the Corresponding author: kareema_kadim@yahoo.com
**
E-mail of the Corresponding author: ahmad_fadhil_han@Yahoo.com
Abstract
Since the widely using of the weighted distribution in many fields of real life such various areas including
medicine, ecology, reliability, and so on, then we try to shed light and record our contribution in this field thru
the research. Derivation Double Weighted distribution and Double Weighted Exponential distribution with some
of statistical properties is discussed in this paper.
Keywords: Weighted distribution, Double Weighted distribution, Double Weighted Exponential distribution.
1. Introduction
Weighted distribution provides an approach to dealing with model specification and data interpretation
problems. Fisher (1934) and Rao (1965) introduced and unified the concept of weighted distribution. Fisher
(1934) studied on how methods of ascertainment can influence the form of distribution of recorded observations
and then Rao (1965) introduced and formulated it in general terms in connection with modeling statistical data
where the usual practice of using standard distributions for the purpose was not found to be appropriate. In Rao' s
paper, he identified various situations that can be modeled by weighted distributions. These situations refer to
instances where the recorded observations cannot be considered as a random sample from the original
distributions. This may occur due to non-observability of some events or damage caused to the original
observation resulting in a reduced value, or adoption of a sampling procedure which gives unequal chances to
the units in the original.
Weighted distributions occur frequently in research related to reliability bio-medicine, ecology and branching
processes can be seen in Patil and Rao (1978), Gupta and Kirmani(1990), Gupta and Keating(1985), Oluyede
(1999) and in references there in. Within the context of cell kinetics and the early detection of disease, Zelen
(1974) introduced weighted distributions to represent twhat he broadly perceived as length-biased sampling
(introduced earlier in Cox, D.R. (1962)). For additional and important results on weighted distributions, see Rao
(1997), Patil and Ord(1997), Zelen and Feinleib (1969), see El-Shaarawi and Walter (2002) for application
examples for weighted distribution, and there are many researches for weighted distribution as, Priyadarshani
(2011) introduced a new class of weighted generalized gamma distribution and related distribution, theoretical
properties of the generalized gamma model, Jing (2010) introduced the weighted inverse Weibull distribution
and beta-inverse Weibull distribution, theoretical properties of them, Castillo and Perez-Casany (1998)
introduced new exponential families, that come from the concept of weighted distribution, that include and
generalize the poisson distribution, Shaban and Boudrissa (2000) have shown that the length-biased version of
the Weibull distribution known as Weibull Length-biased (WLB) distributin is unimodal throughout examining
its shape, with other properties, Das and Roy (2011) discussed the length-biased Weighted Generalized Rayleigh
distribution with its properties, also they are develop the length-biased from of the weighted Weibull distribution
see Das and Roy (2011). On Some Length-Biased Weighted Weibull Distribution, Patil and Ord (1976),
introduced the concept of size-biased sampling and weighted distributions by identifying some of the situations
where the underlying models retain their form. For more important results of weighted distribution you can see
also (Oluyede and George (2000), Ghitany and Al-Mutairi (2008), Ahmed ,Reshi and Mir (2013), Broderick X.
S., Oluyede and Pararai (2012), Oluyede and Terbeche M (2007)).
Suppose is a non-negative random variable with its pdf , then the pdf of the weighted random variable
is given by:
(1)
Where be a non-negative weight function and .

2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
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125
When we use weighted distributions as a tool in the selection of suitable models for observed data is the choice
of the weight function that fits the data. Depending upon the choice of weight function , we have different
weighted models. For example, when , the resulting distribution is called length-biased. In this case,
the pdf of a length-biased (rv) is defined as
(2)
Where .More generally, when , then the resulting distribution is called
size-biased. This type of sampling is a generalization of length-biased sampling and majority of the literature is
centered on this weight function. Denoting , distribution of the size-biased (rv) of order is
specified by the pdf
(3)
Clearly, when , (1) reduces to the pdf of a length-biased (rv).
In this paper we present the Double Weighted distribution (DWD), taking one type of weighted function,
, and using the exponential distribution as original distribution, then we derive the pdf, cdf, and some
other useful distributional properties.
2. Double Weighted Distribution
Definition(2.1): The Double Weighted distribution (DWD) is given by:
(4)
Where ∫
And the first weight is and the second is , depend on the original distribution .
3. Double Weighted Exponential Distribution
Consider the first weight function and the pdf of Exponential distribution given by :-
and
And ∫ ∫ ( )
Then the pdf of Double Weighted Exponential distribution (DWED) is:
( ), (5)
The cdf of DWED is:
∫ ( )
[ ( ( ) )] (6)
Using ∫ and
∫

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4. The Shape
The shape of the density function given in (5) can be clarified by studying this function defined over the
positive real line [0, ∞) and the behavior of its derivative as follows:
4.1 Limit and Mode of the function
Note that the limits of the Density function given in (5) is as follow:-
( ) (7)
( ) (8)
Since and ( )
From these limits, we conclude the pdf of DWED when has one mode say as follow:
Note that ( ) ( ) (9)
Differentiating equation (9) with respect to , we obtain
The mode of the Double Weighted Exponential distribution is obtained by solving the following nonlinear
equation with respect to
(10)
The mode of Double Weighted Exponential distribution is given by Table 1.
The reliability function, hazard function and reverse hazard function of DWED are respectively given by:
[ ( ( ))] (11)
( )
[ ]
(12)
( )
( ) [ ]
(13)
5. Moment Of DWED
The moment of DWED is given by:
[ ]
[ ]
(14)
Where [ ]
Proof:
Let

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Then ∫ ∫
∫ And ∫
Then ∫ ( )
[∫ ∫ ]
[ ] [ ]
[ ]
[ ]
Then from equation (14) we can find mean, variance, coefficient of variation, skewness and kurtosis as follow:
Mean:
[ ]
[ ]
(15)
Variance:
[ ][ ] [ ]
[ ]
(16)
Coefficient of Variation:
[ ( )( ) ( ) ] [ ]
(17)
Coefficient of skewness:
[ ]
[ ]
(18)
Coefficient of kurtosis:
[ ]
(19)
Table 2 shows the mode, mean, standard deviation (STD), coefficient of variation , coefficient of skewness
and coefficient of kurtosis with some values of the parameters . Moment Generating
Function Of DWED
The moment generating function of DWED is given by:
∫ ( )
[∫ ∫ ]
Let

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ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
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So ∫ [ ]
∫ [ ]
And ∫ , then
[ [ ]
] (20)
6. Fisher Information Of DWIWD
The Fisher information (FI) that contains about the parameters are obtained as follows:
Note that
[ ] [ ] (21)
Differentiating (21) with respect to ,we get
(22)
[ ]
[ ] [ ]
(23)
[ ]
[ ]
[ ]
∫
Let
So ∫ ∫ [ ] , we have
∑
( )
( )
(24)
Then ∫ ∫ [ ] ∑
( )
( )
∑
( )
( )
∫ [ ]
Let
Then ∫ ∑
( )
( )
∫ [ ]
∑
( )
( )
∑ ∫ [ ]
( )
∑ ∑
( )
[ ]
[ ]
[ ]
( )
∑ ∑
( )
,(25)

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Now differentiating (21) with respect to , we get
(26)
( )
, so
[ ] ∫
[ ]
∫ [ ]
[ ]
∑
( )
( )
∫ [ ]
[ ]
∑
( )
( )
∫ [ ]
[ ]
∑ ∑
( )
( )
[ ]
∑ ∑
( )
( )
[ ] [ ]
∑ ∑
( )
( )
(27)
Now differentiating (26) with respect to , we get
( )
(28)
[ ] ∫ ∫
Let , then
∫ ∫ [ ]
,
∫ [ ]
∫ [ ]
By the same way above we obtain
∫ [ ]
∑ ∑
( )
( )
[ ] [ ]
∑ ∑
( )
( ) [ ]
(29)
And [ ] [ ] (30)
Now, the Fisher Information Matrix (FIM) for DWED is given by:

7. Mathematical Theory and Modeling www.iiste.org
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Vol.3, No.5, 2013
130
(
[ ] [ ]
[ ] [ ]
) (31)
7. Estimation Of Parameters In The DWED
Method of moment and maximum likelihood are used to estimate the parameters of DWED.
8.1 Method Of Moment Estimators
The method of moments in general provides estimators which are consistent but not as efficient as the
Maximum Likelihood ones. They are often used because they lead to very simple computations. If follows
DWED with parameters and , then the moment of , say is given as
[ ]
[ ]
For the case and let be an independent sample from the DWIWD we obtain the first sample
moment as follow
[ ]
[ ]
∑ (32)
The same way above we can find the estimate for parameter (or ) when (or ) is known as follow
̂ [ ]
[ ]
known (33)
And
̂
√( )
known (34)
We note that if we want to estimate any one of them where the other parameter is unknown, can be obtained by
numerical methods.
8.2 Maximum Likelihood Estimators
This is the best known, most widely used, and most important of the methods of estimation. All that is done
is to write down the likelihood function , and then find the value ̂ of which maximizes . The
log-likelihood function based on the random sample is given by:
[ ] ∑ ∑ ∑
(35)
which admits the partial derivatives
∑ ∑ (36)
And ∑ (37)
Equating these equations to zero, then we get
∑ ∑ (38)

8. Mathematical Theory and Modeling www.iiste.org
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∑ (39)
To find out the maximum likelihood estimators of we have to solve the above system of nonlinear
equations (38)-(39) with respect to . As it seems, this system has no closed form solution in . Then we
have to use a numerical technique method, such as Newton-Raphson method (see Adi (1966)), to obtain the
solution. An example below show that.
Example(1):
Table 3 represents the total annual rainfall (mm) in the province of Babylon, and for the period of (1998-
2009) see (Environmental Statistics in Iraq Report, (2009)). The data can be modeled by Double Weighted
Exponential distribution and also we estimate the parameters and using Newton Raphson method (see Adi
(1966)), beginning with the initial estimates ̂ and ̂ . Then the estimate values are: ̂
and ̂ , for 6 iterated.
8. Conclusions
We can develop the weighted distribution into Derivation Double weighted distribution (DWD), like Double
Weighted distribution, Double Weighted Exponential distribution. Therefore, We can discuss some of statistical
properties on them.
References
Adi Ben-Israel, (1966), A Newton-Raphson Method For The Solution Of Systems Of Equations, Journal of
Mathematical analysis and applications. Vol. 15, No. 2.
Ahmed A. ,Reshi J. A. and Mir K. A. (2013). Structural Properties of size biased Gamma distribution,
IOSR Jornal of Mathematics, Vol. 5, Issue 2, pp 55-61.
Broderick X. S., Oluyede B. O. and Pararai M., (2012). Theoretical Properties of Weighted Generalized
Rayleigh and Related Distributions, Theortical Mathematics & Applications, Vol. 2, no.2, 2012, 45-62.
Castillo, J.D. and Casany, M. P., (1998). Weighted Poisson Distributions and Underdispersion Situations,
Ann. Inst. Statist. Math. Vol. 50, No. 3. 567-585.
Cox, D.R. (1962). Renewal Theory, Barnes & Noble, New York.
Das K. K. and Roy T. D. (2011). On Some Length-Biased Weighted Weibull Distribution, Pelagia Research
Library,Advances in Applied Science Research, , 2 (5):465-475. ISSN: 0976-8610, USA.
Das, K. K. and Roy, T. D. (2011). Applicability of Length Biased Weighted Generalized Rayleigh
distribution, Advances in Applied Science Research, , 2 (4): 320-327.
El-Shaarawi, Abdel .and Walter W. Piegorsch, (2002). Weighted distribution, G. P. Patil, Vol. 4, pp 2369-
2377.
Environmental Statistics in Iraq Report, (2009). Republic of Iraq, Ministry of Planning, Central
Organization For Statistics COSIT.
Fisher, R.A. (1934). The effects of methods of ascertainment upon the estimation of frequencies, The Annals
of Eugenics 6, 13–25.
Ghitany, M. E. and Al-Mutairi, D. K. (2008). Size-biased Poisson-Lindley distribution and its application,
METRON-International Journal of Statistics. Vol. LXVI, n. 3, pp. 299-311.
Gupta, R. C., and Keating, J. P. (1985). Relations for Reliability Measures under Length Biased Sampling,
Scan. J. Statist., 13, 49-56,
Gupta, R. C., and Kirmani, S. N. U. A. (1990). The Role of Weighted Distributions in Stochastic Modeling,
Commun. Statist., 19(9), 3147-3162.

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G.P. Patil and J.K. Ord, (1997). Weighted distributions, in Encyclopedia of Bio-statistics, P. Armitage and
T. Colton, eds, 6, Wiley, Chichester, 4735-4738.
JING, X. K. (2010). Weighted Inverse Weibull and Beta-Inverse Weibull distribution, 2009 Mathematics
Subject Classification: 62N05, 62B10
M. Zelen and M. Feinleib, On the Theory of Chronic Diseases, Biometrika, 56, (1969), 601-614.
Oluyede, B. O. (1999). On Inequalities and Selection of Experiments for Length-Biased Distributions,
Probability in the Engineering and Informational Sciences, 13, 169-185.
Oluyede, B. O. and George, E. O. (2000). On Stochastic Inequalities and Comparisons of Reliability
Measures for Weighted Distributions. Mathematical problems in Engineering, Vol. 8, pp. 1-13.
Oluyede B. O. and Terbeche M.,(2007). On Energy and Expected Uncertainty Measures in Weighted
Distributions, International Mathematical Forum, 2, no. 20, 947-956.
Patil, G.P. &Rao, C.R. (1978). Weighted distributions and size-biased sampling with applications to wildlife
populations and human families, Biometrics34, 179–184.
Patil, G. P. and Ord,J. K.,(1976). On size-Biased Samling and Related Form-Invariant Weighted
distribution, Sankhya: The Indian Journal Of Statistics, Vol. 38, Series B, Pt. 1, pp. 48-61.
Priyadarshani, H. A. (2011). Statistical properties of weighted Generalized Gamma distribution,
Mathematics Subject Classification: 62N05, 62B10.
Rao, C.R. (1965). On discrete distributions arising out of methods of ascertainment, in Classical and
Contagious Discrete Distributions, G.P. Patil, ed., Pergamon Press and Statistical Publishing Society,
Calcutta, pp. 320–332.
Rao, C.R. (1997).Statistics And Truth Putting Chance to Work, World Scientific Publishing Co. Pte.
Ltd.Singapore.
Shaban, S. A. and Boudrissa, N. A. (2000). The Weibull Length Biased distribution properties and
Estimation, MSC Primary: 60E05, secondary: 62E15, 62F10 and 62F15
Zelen, M. (1974). Problems in cell kinetics and the early detection of disease, in Reliability and Biometry, F.
Proschan& R.J. Serfling, eds, SIAM, Philadelphia, pp. 701–706.
Dr. Kareema A. K,, assistant professor, Babylon University for Pure Mathematics Dept.. Her
:1989/1990 3) the Bachelor's degree in Statistics: 1980/1981 4) the Bachelor's degree in English
Language: 2007general specialization is statistics and specific specialization is Mathematics Statistics. Her
interesting researches are : Mathematics Statistics, Time Series, and Sampling that are poblished in
international and local Journals. The certificates and the dates the obtained
1) the Doctor degree in Mathematical Statistics: 30/12/2000 2) the Master degree
Statistics:1989/1990 3) the Bachelor's degree in Statistics: 1980/19814) the Bachelor's degree in
English Language: 2007/2008

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Table 1: mode of DWED.
Figure 1: the pdf of DWED for specified values of and .
Figure 1: the pdf of DWED for specified values of and .
Table 2 the mode, mean, standard deviation (STD), coefficient of variation , coefficient of skewness
and coefficient of kurtosis with some values of the parameters of DWED.
Mode Mean STD VAR
1 2
3
6.
2
12
1.2298
1.1207
1.0118
1.0000
2.1667
2.1000
2.1208
2.0110
1.4044
1.4018
1.4068
1.4112
1.9722
1.9650
1.9791
1.9914
0.4583
0.4720
0.4891
0.4962
3.9808
4.0401
4.0450
4.0218
6.0268
6.0802
6.0594
6.0254
2 2
3
6.
2
12
0.6149
0.5603
0.5059
0.5000
1.0833
1.0500
1.0169
1.0055
0.7022
0.7009
0.7034
0.7056
0.4931
0.4913
0.4948
0.4979
0.4583
0.4720
0.4891
0.4962
3.9808
4.0401
4.0450
4.0218
6.0268
6.0802
6.0594
6.0254
5 2
3
6.
2
12
0.2459
0.2241
0.2023
0.2000
0.4333
0.4200
0.4068
0.4022
0.2809
0.2804
0.2814
0.2822
0.0789
0.0786
0.0792
0.0797
0.4583
0.4720
0.4891
0.4962
3.9808
4.0401
4.0450
4.0218
6.0268
6.0802
6.0594
6.0254
8.
7
2
3
6.
2
12
0.1413
0.1288
0.1163
0.1149
0.2490
0.2414
0.2338
0.2311
0.1614
0.1611
0.1617
0.1622
0.0261
0.0260
0.0261
0.0263
0.4583
0.4720
0.4891
0.4962
3.9808
4.0401
4.0450
4.0218
6.0268
6.0802
6.0594
6.0254
c Mode
2
4
6
8.2
9.7
1
2
5
7.6
2
1.2298
0.6149
0.2459
0.1618
0.5308
0.5069
0.5011
0.5002

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From Table 2, we note that the mode decreases at fixed and change
. And so the mean, except at . The STD and VAR take valuable oscillatory between decreasing at
with . we note that , and do not depend on , but only on . That is, the
increases at all values of . While the and takes valuable oscillatory between increasing into
decreasing.
The Figure follow shows the plots of , and for DWED.
Figure 2: Plots of , and for DWED.
Figure 2 plot of , and
From Figure 2 (1, 2, 3), we note that , and do not depend on the parameter . In -1-, the
behaves monotonically increases when increasing. The relationship between and is shown in -2-, such
that the behaves monotonically increases when increasing until it reaches the maximum value of is
4.0525 for 4.3 and then decreases to settle down in value for the larger value of . From our
calculations it's clear that , then ( Mean Mode), see table 2 and the pdf of DWED is skewed to the
right. The relationship between and is shown in -3-, such that the behaves monotonically
increases when increasing until it reaches the maximum value of is 6.0833 for 3.5 and then
decreases to settle down in value for the larger value of . Since in our calculations it's clear that
then the pdf of DWED shape is more peaked than the Normal pdf.
Table 3: Total annual rainfall (mm) in the province of Babylon, and for the period of (1998-2009)
1998 1999 2000 2001 2003 2003 2004 2005 2006 2007 2008 2009
95.8 65.3 85.3 81.3 102.8 134.5 71.1 73.2 170.3 41.0 51.8 52.4